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hep-ph/0405110

CERN–TH/2004-069, UMN–TH–2307/04, FTPI–MINN–04/18

Very Constrained Minimal Supersymmetric Standard Models

John Ellis1, Keith A. Olive2, Yudi Santoso2 and Vassilis C. Spanos2

1TH Division, CERN, Geneva, Switzerland

2William I. Fine Theoretical Physics Institute,

University of Minnesota, Minneapolis, MN 55455, USA

Abstract

We consider very constrained versions of the minimal supersymmetric extension of the

Standard Model (VCMSSMs) which, in addition to constraining the scalar masses m0 and

gaugino masses m1/2 to be universal at some input scale, impose relations between the

trilinear and bilinear soft supersymmetry breaking parameters A0 and B0. These relations

may be linear, as in simple minimal supergravity models, or nonlinear, as in the Giudice-

Masiero mechanism for generating the Higgs-mixing µ term. We discuss the application of

the electroweak vacuum conditions in VCMSSMs, which may be used to make a prediction

for tanβ as a function of m0 and m1/2 that is usually unique. We baseline the discussion

of the parameter spaces allowed in VCMSSMs by updating the parameter space allowed

in the CMSSM for fixed values of tanβ with no relation between A0 and B0 assumed a

priori, displaying contours of B0 for a fixed input value of A0, incorporating the latest

CDF/D0 measurement of mt and the latest BNL measurement of gµ−2. We emphasize that

phenomenological studies of the CMSSM are frequently not applicable to specific VCMSSMs,

notably those based on minimal supergravity, which require m0 = m3/2 as well as A0 =

B0 + m0. We then display (m1/2, m0) planes for selected VCMSSMs, treating in a unified

way the parameter regions where either a neutralino or the gravitino is the LSP. In particular,

we examine in detail the allowed parameter space for the Giudice-Masiero model.

CERN–TH/2004-069

May 2004

1 Introduction

Supersymmetry is one of the most appealing extensions of the Standard Model (SM), for

many reasons including the hierarchy problem, its necessity in string theory, unification

of the SM gauge couplings, the suggestion of a light Higgs boson, the possibility that the

astrophysical cold dark matter might be provided by the lightest supersymmetric particle

(LSP) and (just possibly) the anomalous magnetic moment of the muon, gµ − 2. However,

supersymmetry is a general framework that accommodates many new degrees of freedom.

The simplest possible realization of supersymmetry is the minimal supersymmetric extension

of the SM (MSSM). Four types of supersymmetry-breaking parameters appear in the MSSM:

scalar masses m0, gaugino masses m1/2, trilinear couplings A and a bilinear coupling B in

the Higgs sector. In the MSSM alone, the number of free parameters associated with soft

supersymmetry breaking exceeds 100, unless one assumes some degree of universality for

the sparticles with different quantum numbers and flavours. In phenomenological studies

of supersymmetry, the values of m0 for the different sflavours are often constrained to be

universal at some input GUT scale, as are the values of m1/2 for the different SM gauge group

factors, and the A parameters corresponding to different SM Yukawa couplings, a framework

often called the CMSSM.

One may go even further, and assume some relation(s) between the parameters m0, m1/2,

A, B and the gravitino mass m3/2. In particular, many very constrained versions of the

MSSM (VCMSSMs) derive or postulate relations between the A and B parameters, which

we parametrize as A ≡ Am0, B ≡ Bm0. These relations may be linear: for example, generic

minimal supergravity models predict that B = A − 1 [1, 2] as well as m0 = m3/2, and the

simplest Polonyi model [3] of supersymmetry breaking further predicts that |A| = 3−√

3 [4].

On the other hand, a prominent example of a nonlinear relation is

B =2A − 3

A − 3, (1)

which appears in the Giudice-Masiero mechanism [5] for generating the µ term.

In the CMSSM, one may regard m0, m1/2 and A as independent parameters, and use

the two electroweak vacuum conditions resulting from the specification of mZ and the ratio

of Higgs vacuum expectation values, tan β, to fix |µ| and the pseudoscalar Higgs mass mA,

which is equivalent to fixing B. As we show in this paper with some explicit examples, the

value of B that results for any given choice of m0, m1/2 and A may not correspond to any

plausible theoretical model. Conversely, in a VCMSSM where B is fixed in terms of A, one

can use the electroweak vacuum conditions to predict tanβ as a function of m0, m1/2 and A.

1

In a previous paper [6], we demonstrated this type of prediction for a few specific VCMSSMs

with linear relations between A and B, including minimal supergravity, with the simplest

Polonyi model as a special case.

In this paper, we extend the previous discussion to include the Giudice-Masiero model.

In this case, in addition to the relation (1) between B and A, the value of µ is in principle

also predicted as∣

∣

∣

∣

µ

m0

∣

∣

∣

∣

=

∣

∣

∣

∣

∣

λA − 3√

3

∣

∣

∣

∣

∣

(2)

where λ/MP l is the coupling between a hidden sector superfield and the two Higgs superfields.

The value of λ is presumably not completely arbitrary: for example, one should probably

require |λ| <∼ O(1). This bound on |λ| in turn imposes a range on the ratio µ/m0 for a given

A. Since the value of |µ| is an output quantity in our approach to VCMSSMs, one must

check that |λ| is not very large, which could in principle restrict the ranges of the input

parameters.

The first step in this paper is to discuss the application of the electroweak vacuum

conditions. In principle, more than one value of tanβ might be consistent with a given

VCMSSM for some specific choice of (m0, m1/2) [7]. In practice, over large regions of the

(m0, m1/2) we find only one solution for tanβ, as we explain in some detail. We also discuss

the renormalization of the input relation between A and B in a generic VCMSSM, including

the relation between the input and electroweak-scale values of B and the one-loop threshold

corrections at the electroweak scale.

The second step is to update previous analyses of the CMSSM, including some updates

in calculations of the supersymmetric particle spectrum, as well as the latest information on

mt, gµ − 2, b → sγ and Bs → µ+µ−. We demonstrate that the values of B required in the

CMSSM for generic values of tan β and A do not fit within favoured VCMSSM frameworks,

such as those based on minimal supergravity or the Giudice-Masiero model.

We then discuss the (m0, m1/2) planes for some specific VCMSSMs, taking into account

the fact that minimal supergravity models predict that m0 = m3/2 before renormalization,

which is not necessarily the case in a generic CMSSM. This relation enables one to delineate

the regions where the LSP is the lightest neutralino χ, the lighter τ or the gravitino G.

We present unified descriptions of the χ and G LSP regions for some specific VCMSSMs,

incorporating the constraints on decays of the next-to-lightest supersymmetric particle (NSP)

into a gravitino LSP that are imposed by concordance between the Big-Bang nucleosynthesis

(BBN) and cosmological microwave background (CMB) determinations of the baryon-to-

entropy ratio [8, 9, 10, 11]. Finally, we discuss the Giudice-Masiero model in more detail,

2

finding that the implied values of |λ| in allowed regions of parameter space are generally

>∼ O(1), particularly in the gravitino LSP region.

2 Summary of Models of Supersymmetry Breaking

As discussed in [6], we assume an N = 1 supergravity framework, interpreted as a low-

energy effective field theory. In minimal supergravity models, the Kahler function K that

describes the kinetic terms for the chiral supermultiplets Φ ≡ (ζ, φ), where the ζ represent

hidden-sector fields and the φi observable-sector fields, has the form K = Σi|Φi|2. We denote

derivatives of K with respect to the chiral superfields by Ki ≡ ∂K/∂φi, etc. In the minimal

supergravity case, we have Ki = φi∗ + W i/W , Ki = φi + W ∗i /W ∗, and (K−1)j

i = δji , and the

resulting scalar potential is (in units where the Planck mass is unity)

V (φ, φ∗) = eφiφi∗

[

|W i + φi∗W |2 − 3|W |2]

. (3)

It is then apparent that the soft supersymmetry-breaking scalar masses m0 are universal at

the input GUT scale, with [1]

m20 = m2

3/2, (4)

where m3/2 is the gravitino mass and we assume that the tree-level cosmological constant

vanishes. If we further assume that the superpotential W (Φ) may be separated into pieces

f and g that are functions only of observable-sector fields φi and hidden-sector fields ζ ,

respectively, then the soft supersymmetry-breaking trilinear terms A0 and bilinear terms B0

are also universal, and are related by [1]

B0 = A0 − m3/2, (5)

so that

B = A − 1, (6)

which is one of the principal options we studied in [6] and discuss further below.

The simplest model for local supersymmetry breaking in minimal supergravity [3] has

just one additional chiral multiplet ζ in addition to the observable matter fields φi, with a

superpotential that is separable in this so-called Polonyi field and the observable fields φi:

W = f(φ) + g(ζ). It takes the simple form

g(ζ) = ν(ζ + β), (7)

3

where we impose |β| = 2−√

3 to ensure that the cosmological constant vanishes. Assuming β

to be positive, and using 〈ζ〉 =√

3−1, we have [4] the universal soft trilinear supersymmetry-

breaking terms

A = (3 −√

3)m3/2, (8)

and universal bilinear soft supersymmetry-breaking terms

B = (2 −√

3)m3/2, (9)

whose consequences we explored in [6] and discuss further below.

In the simplest version of the Giudice-Masiero (GM) mechanism [5], in addition to mini-

mal supergravity kinetic terms in the observable and hidden sectors, and a separable super-

potential W = f + g, one postulates a coupling

K(φ, ζ) ∋ λζ†H1H2, (10)

where H1,2 are the two Higgs supermultiplets in the MSSM. Assuming that the cosmological

constant vanishes, the term (10) generates a Higgs mixing term (2). This mechanism also

yields the nonlinear relation between B and A given in (1), whose consequences we explore

below.

As already remarked, minimal supergravity models predict a relation (4) between m0 and

the gravitino mass, which is not necessarily true in the generic CMSSM. This relation enables

us to delineate the regions of VCMSSM parameter space where the LSP is a neutralino, the

lighter τ or the gravitino. The astrophysical and cosmological constraints on gravitino dark

matter have been recently re-examined [8, 9], taking also into account the constraints on

decays of the next-to-lightest supersymmetric particle (NSP) arising from comparing the

BBN and CMB constraints on the baryon-to-entropy ratio [10, 11]. In our later discussions

of VCMSSMs, we give unified treatments of the parts of (m1/2, m0) planes where the LSP is

a neutralino, the lighter τ and the gravitino.

3 The Electroweak Vacuum in VCMSSMs

In the general CMSSM, we start with the following set of input parameters defined at

the GUT scale: m1/2, m0, A0, B0 and the Higgs mixing parameter µ0. At tree level, the

electroweak vacuum is specified by the following two conditions:

m2Z =

2(m21 + µ2 − (m2

2 + µ2) tan2 β)

(tan2 β − 1), (11)

sin 2β = −2Bµ/(m21 + m2

2 + 2µ2), (12)

4

and the pseudoscalar neutral Higgs mass mA is determined by

m2A = m2

1 + m22 + 2µ2, (13)

where m1 and m2 are the soft supersymmetry-breaking masses for the two Higgs dou-

blets at the electroweak scale. These as well as µ and B are assumed to be evaluated

by renormalization-group equation (RGE) running from the input values. One may, alter-

natively, solve for µ and B in terms of mZ and tan β:

µ2 =m2

1 − m22 tan2 β + 1

2m2

Z(1 − tan2 β) + ∆(1)

µ

tan2 β − 1 + ∆(2)µ

Bµ = −1

2(m2

1 + m22 + 2µ2) sin 2β + ∆B (14)

where we have now included the loop corrections ∆B and ∆(1,2)µ required to relate the

RGE values to the corresponding quantities evaluated at mZ [12, 13, 14], and here m1,2 ≡m1,2(mZ) 1. In most treatments of the CMSSM, m1/2, m0, A0 and tanβ are taken as inputs,

and the conditions (14) are used to determine µ, B and the CP-odd Higgs mass mA.

As discussed in [6], in a VCMSSM where B is determined in advance in terms of A, it is

convenient to use the electroweak vacuum conditions (14) to determine tanβ as a function of

m0 and m1/2 for some input value of A. However, since ∆µ depends on tan β, and ∆B depends

on both µ and tan β in a nonlinear way, it is not possible to write down an analytical solution

for tanβ. Moreover, it was shown in [7] using an RGE-improved tree-level calculation for

tan β in the minimal supergravity model that there may be up to three possible solutions

for tan β for any given choices of m1/2, m0, and A0. We remarked previously [6] that we

typically find just one solution with a moderately low value of tan β, that multiple solutions

exist only for mt < 153 GeV, and that B always increases monotonically with tanβ over the

range 5 < tanβ < 55 in our calculations. Thus, a given value of m1/2, m0, A0 and sgn(µ)

always corresponds, in our analysis, to a definite value for tan β. Since this is important for

our treatment of VCMSSMs, we now illustrate this point in more detail.

We show in Fig. 1 some examples of the necessary input values of B0 as functions of tanβ,

for four representative choices of (m0, m1/2) and µ > 0. We use mt = 178 GeV as suggested

by the latest CDF and D0 results [15]. We see that B0 generally increases monotonically

for all positive values of A0, and also for some negative values of A0. This is also true for

µ < 0, as seen in Fig. 2. These observations immediately imply that, in any VCMSSM that

predicts a unique value of B0 for a given value of A0, there will be (at most) a unique value

of tan β where the VCMSSM relation is obeyed.

5

1 10 20 30 40 50 59-5

-4

-3

-2

-1

0

1

2

3

4

5

B0

(103

GeV

)

tan β

m0 = 1000 GeV, m1/2 = 200 GeV, µ > 0

A= 2^

mt=178 GeV

A= -2^

0

1

1 10 20 30 40 50 59-5

-4

-3

-2

-1

0

1

2

3

4

5

B0

(103

GeV

)tan β

m0 = 1000 GeV, m1/2 = 1000 GeV, µ > 0

A= 4^

mt=178 GeV

A= -8^

-6

-4

-2

0

1 2

1 10 20 30 40 50 59-5

-4

-3

-2

-1

0

1

2

3

4

5

B0

(103

GeV

)

tan β

m0 = 200 GeV, m1/2 = 200 GeV, µ > 0

A= 4^

mt=178 GeV

1 0

-2-4

2

A= -10^

1 10 20 30 40 50 59-5

-4

-3

-2

-1

0

1

2

3

4

5

B0

(103

GeV

)

tan β

m0 = 200 GeV, m1/2 = 1000 GeV, µ > 0

A= -40^

mt=178 GeV

12410

0-2

-4

-10

-20

-30

A= 20^

Figure 1: Values of B0 as functions of tanβ for µ > 0 and (m1/2, m0) = (a) (200, 1000) GeV,

(b) (1000, 1000) GeV, (c) (200, 200) GeV and (d) (1000, 200) GeV. Solutions for B = A− 1case are denoted by small circles, which are connected by dashed lines. Solutions in the caseof the Giudice-Masiero mechanism are denoted by small squares, connected by dot-dashedlines when possible.

6

1 10 20 30 40-5

-4

-3

-2

-1

0

1

2

3

4

5

B0

(103

GeV

)

tan β

m0 = 1000 GeV, m1/2 = 200 GeV, µ < 0

A= 2^

mt=178 GeV

A= -2^

1

0

1 10 20 30 40-5

-4

-3

-2

-1

0

1

2

3

4

5

B0

(103

GeV

)

tan β

m0 = 1000 GeV, m1/2 = 1000 GeV, µ < 0

A= 4^

mt=178 GeV

1

0

-2

-4

-6

2

A= -8^

1 10 20 30 40-5

-4

-3

-2

-1

0

1

2

3

4

5

A= 4^

A= -10^

-4-2

012

m0 = 200 GeV, m1/2 = 200 GeV, µ < 0

mt=178 GeV

B0

(103

GeV

)

tan β1 10 20 30 40

-5

-4

-3

-2

-1

0

1

2

3

4

5

B0

(103

GeV

)

tan β

m0 = 200 GeV, m1/2 = 1000 GeV, µ < 0

A= 20^

mt=178 GeV

01

2 4 10

-2

-4

-10

-20

-30

A= -40^

Figure 2: As in Fig. 1, but for µ < 0.

7

We note, however, that there are some particular negative values of A0 for which the

required value of B0, after rising when tan β is small, decreases slightly at large tan β. This

raises the possibility that there might be two allowed values of tan β in some restricted set

of VCMSSMs. One example is when m1/2 = 200 GeV, m0 = 1000 GeV, µ > 0, A ≃ −2 and

B ≃ −1.5, as seen in panel (a) of Fig. 1, and there are some other examples in other panels

of Figs. 1 and 2. However, in practice, such multiple solutions do not exist in the specific

VCMSSMs that we study in this paper.

To illustrate this more explicitly, we indicate by small circles in Figs. 1 and 2 the values

of tan β where the minimal supergravity condition B = A − 1 is satisfied, for a few specific

values of A, and we join these points by dashed lines. For each value of A there is clearly

only one consistent choice of tan β for any given choice of (m1/2, m0). We also show solutions

for the Giudice-Masiero mechanism case, indicated by small squares.

Note that we do not obtain solutions for B for all choices of A. For example, in Fig. 1a,

we show solutions only for A = 0, 1 and 2 for B = A− 1 and A = 1 for the Giudice-Masiero

model. In the minimal supergravity case, when A is reduced, B is also reduced driving

the solution to smaller values of tan β. Very quickly these solutions drop below tan β = 2

and, below tanβ ∼ 1.7, the RGEs do not provide solutions to the sparticle spectra due to

a divergence in the top quark Yukawa coupling at the unification scale. Similarly when A

is large, the solution is driven to very large values of tanβ where again no solutions to the

RGEs are found. In the case of the GM model, the slope of B0 vs tanβ is very small, and

small changes in A lead to large changes in tanβ. Note also that in the GM model, there are

often two branches of solutions which are disconnected. This is seen for example in Figs. 1d

and 2d. This is due to the relation (1) which separates solutions at A = 3.

In order to have an analytical feel for the solutions for B0 shown in Fig. 1 and 2, we show

in Fig. 3 the values of B0 at the input GUT scale, the tree-level values at the electroweak

scale and the full values of B(MW ) as functions of tan β, (a) for µ > 0 and (b) for µ < 0,

in both cases for (m1/2, m0) = (200, 200) GeV. The tree-level value of B at the electroweak

scale is defined as

Btree ≡ (m21 + m2

2 + 2µ2) sin 2β

−2µ, (15)

and as one can see Btree tends to 0 as tan β is increased. The ‘full’ values are calculated

including one-loop electroweak threshold corrections, and B0 is then the result of running

the RGEs from the weak scale to the unification scale. In the µ > 0 case, we see in Fig. 3

that B0 is systematically larger than the tree-level value of B(MW ), which is in turn larger

1As observed in [6], comparisons [16] with ISASUGRA [17] show that our procedure of minimizing the Higgspotential at the weak scale gives very similar spectra, also at large tanβ and in the focus-point region.

8

1 10 20 30 40 50 59-500

-400

-300

-200

-100

0

100

200

300

400

500

B (

GeV

)

tan β

m0 = 200 GeV, m1/2 = 200, µ > 0, A=1^

B0

B(MW)

Btree

1 10 20 30 40-500

-400

-300

-200

-100

0

100

200

300

400

500

B (

GeV

)

tan β

m0 = 200 GeV, m1/2 = 200, µ < 0, A=1^

B0

B(MW)

Btree

Figure 3: The values of B0, the electroweak tree-level and full B(MW ), as functions of tanβfor (a) µ > 0 and (b) µ < 0, both for (m1/2, m0) = (200, 200) GeV and A = 1.

than its full value. However, even in this case B(MW ) increases monotonically with tanβ.

The situation is rather different for µ < 0, where we see that the sign of the loop correction

depends on the value of tan β, vanishing for tanβ ≃ 21. As a result, the full value of B(MW )

and hence B0 increase monotonically with tanβ. Had we neglected the 1-loop corrections to

B and ran the RGEs up to the unification scale, we could obtain a non-monotonic solution

for B0 with respect to tan β (for example, a solution with a minimum value of B0) leading to

multiple solutions of tan β for a fixed value of A [18]. Thus, Fig. 3 indicates the importance

of the loop correction in determining the number of solutions.

4 Updated Constraints on the CMSSM

The standard LEP constraints and cosmological constraints on the CMSSM have been dis-

cussed previously in many places [19, 20], so we do not discuss them further here, except to

recall that we use the WMAP range 0.094 < Ωχh2 < 0.129 [21] for the relic density of the

LSP, assumed to be the lightest neutralino χ. However, there are three new experimental

developments that we should like to mention. One is the new value mt = 178.0 ± 4.3 GeV

recently reported by the CDF and D0 collaborations [15], another is the evolution in the

possible discrepancy between the experimental measurement of gµ − 2 and the value calcu-

9

lated in the SM, and the other is a recent improved upper limit on the branching ratio for

Bs → µ+µ−.

The new value of mt affects the CMSSM parameter space in three important ways. One

is to alter the calculation of the lightest MSSM Higgs boson mass mh, and hence the lower

limit on m1/2 inferred from the LEP lower limit mh > 114.4 GeV. For example, in panel (a)

of Fig. 4 for tan β = 10 and µ > 0, the lower limit on m1/2 is reduced by about 50 GeV when

one increases mt from 175 GeV to the value of 178 GeV shown here. A second effect is to

alter the calculation of the rapid-annihilation funnels shown in panels (c) and (d) of Fig. 4

for tanβ = 35 and µ < 0 and for tan β = 50 and µ > 0, respectively. The sensitivity of these

regions to mt and large tan β was discussed earlier [22] in the context of the observability

of the Higgs boson at hadron colliders. Finally, the larger value of mt increases significantly

the value of m0 where the focus-point region may be found [23]. For example, for tan β = 10

and µ > 0, we now find a focus-point region only for m0>∼ 7 TeV for m1/2

>∼ 250 GeV. We

do not discuss focus points further in this paper.

The BNL gµ − 2 experiment recently announced a new determination using µ− and a

final combined value using all their µ± data [24]. Comparing with the SM calculations of

Davier et al. [25], they quote a discrepancy of aµ ≡ (gµ − 2)/2 with the SM amounting to

δaµ = (27 ± 10) × 10−10 (e+e− data)

= (12 ± 9) × 10−10 (τ data). (16)

Another calculation of the SM contribution to (gµ − 2) using just the e+e− data [26] yielded

a slightly larger discrepancy:

δaµ = (32 ± 10) × 10−10 (e+e− data) (17)

There has subsequently been a new SM calculation of the hadronic vacuum polarization

contribution by de Troconiz and Yndurain [27], who quote

δaµ = (27 ± 8) × 10−10 (e+e− data)

= (19 ± 8) × 10−10 (τ and e+e− data). (18)

However, neither of these evaluations include the recent re-evaluation of the light-by-light

contribution to aµ by Melnikov and Vainshtein [28], which decreases the discrepancy with the

SM by about 4 × 10−10 compared with (18). Therefore, for the purposes of the subsequent

discussion, we show contours corresponding to

δaµ = (15 ± 8) × 10−10. (19)

10

We exhibit this constraint at the 2-σ level, in which case its effect is essentially to exclude the

option µ < 0 but allow most of the (m1/2, m0) plane for µ > 0, apart from a region of small

m1/2 and m0. However, we are well aware that the range (19) is open to question, particularly

in view of the discrepancy between the estimates of the SM contribution based on e+e− and

τ data, and, to a lesser extent, the uncertainty in the light-by-light contribution. Therefore,

we use (19) only as an indication, and by no means a rigid constraint on the parameter space

of the CMSSM or any VCMSSM. In particular, we do not discard the option µ < 0.

Finally, we note that the CDF Collaboration have recently published an improved exper-

imental upper limit on the branching ratio for Bs → µ+µ− [29], namely 5.8×10−7. Since the

branching ratio for this decay ∝ tan6 β in the CMSSM, this constraint is potentially impor-

tant at large tanβ. We find that this constraint is currently still ‘covered’ by the constraints

from b → sγ, mh and gµ − 2, but this situation may change in the near future.

In preparing the (m1/2, m0) planes in Fig. 4 and the subsequent figures, we have updated

our code by making improvements that have impacts principally in the rapid-annihilation

funnels and focus-point regions 2. Their effects are smaller than the other effects mentioned

above.

We show in Fig. 4 the (m1/2, m0) planes for a popular set of CMSSM cases, namely

(a) tanβ = 10, µ > 0, (b) tan β = 10, µ < 0, (c) tan β = 35, µ < 0, and (d) tanβ =

50, µ > 0, all for A = 0 3. In each panel, as well as the ‘standard’ experimental and

cosmological constraints, we have indicated some representative contours of B by (blue)

dashed lines. We see that B >∼ 0 in almost all the (m1/2, m0) planes exhibited, which is

clearly incompatible with the minimal supergravity condition B = A − 1. This exemplifies

the point that parameter choices allowed in the ‘standard’ CMSSM are often not allowed in

favoured VCMSSMs. Specifically, the CMSSM cases shown in Fig. 4 could not be realized

in minimal supergravity: one needs to choose smaller values of tanβ. A similar conclusion

applies to the Giudice-Masiero model, which would require B = 1 for the case A = 0

considered here, although GM solutions are possible for µ < 0 if one discards the gµ − 2

constraint and chooses tan β somewhat above 10.

Fig. 5 shows the corresponding (m1/2, m0) planes for the same choices of tanβ and the

sign of µ, but for A = +0.75. In this case, the laboratory and cosmological constraints are

not greatly different, even in the rapid-annihilation funnel regions 4. Now, however, there are

2Specifically, we now include the full one-loop corrections to mb and mt instead of approximate expres-sions [30], and we correct a minor coding error.

3We note in panels (c) and (d) the appearance of allowed bands above the χ − τ1 coannihilation strips,which are due to rapid ˜τ1τ1 → H annihilation.

4We note again the rapid τ1˜τ1 → H annihilation strips in panels (c) and (d).

11

100 200 300 400 500 600 700 800 900 10000

100

200

300

400

500

600

700

800

1

.

.

m1/2 (GeV)

m0

(GeV

)tan β = 10, µ > 0, A = 0^

mh = 114 GeV

-0.25

-0.25

-0.5

-1 -2

-5

-0.1

mt = 178 GeV

100 200 300 400 500 600 700 800 900 10000

100

200

300

400

500

600

700

800

m1/2 (GeV)m

0 (G

eV)

tan β = 10, µ < 0, A = 0^

mh = 114 GeV

5

21

2/30.5

0.25

0.10

-0.1 mt = 178 GeV

100 1000 20000

1000

2

m1/2 (GeV)

m0

(GeV

)

tan β = 35, µ < 0, A = 0^

mh = 114 GeV

0.1

0.25 0.5

2/3

1

2

5

mt = 178 GeV

2

100 1000 2000 30000

1000

1500

m1/2 (GeV)

m0

(GeV

)

tan β = 50, µ > 0, A = 0^

mh = 114 GeV

mt = 178 GeV

0.1

0.1

0.25

0.5

2/3

15

Figure 4: The (m1/2, m0) planes in the CMSSM for (a) tan β = 10, µ > 0, (b) tan β =

10, µ < 0, (c) tanβ = 35, µ < 0, and (d) tan β = 50, µ > 0, all for A = 0. In eachpanel, the region allowed by the WMAP cosmological constraint 0.094 ≤ Ωχh2 ≤ 0.129 haspale (turquoise) shading. The disallowed region where mτ1 < mχ has dark (red) shading.The regions excluded by b → sγ have medium (green) shading, and those in panels (a,d)that are disfavoured by gµ − 2 at the 2-σ level have very light (yellow) shading with a thin(black) border. The contours mχ± = 104 GeV (mh = 114 GeV) are shown as near-vertical

black dashed (red dot-dashed) lines. In addition, we show several contours of B as (blue)dashed lines. There is no allowed point compatible with the minimal supergravity conditionB = A − 1 or the Giudice-Masiero model in these plots.12

some points where the minimal supergravity condition B = A − 1 is satisfied, as shown by

the intersection of the B = −0.25 line with the WMAP coannihilation region in Fig. 5a. For

this choice of A, the Giudice-Masiero model is also satisfied at a limited number of points,

exemplified by the intersection of the B = 2/3 line with the WMAP funnel region in Fig. 5d.

5 Examples of (m1/2, m0) Planes in VCMSSMs

We now discuss the impacts of the above constraints on some specific VCMSSMs within

the general framework of minimal supergravity, in which B = A − 1. As usual, we display

these constraints in (m1/2, m0) planes. For the reasons discussed earlier, we regard tan β as

a dependent quantity that varies across these planes, rather than being a fixed quantity as

in most CMSSM analyses. Another difference from most CMSSM analyses is that the latter

generally consider only the possibility that the LSP is the lightest neutralino χ, assuming

implicitly that the gravitino mass m3/2 is sufficiently large that the gravitino LSP possibility

can be neglected. However, in minimal supergravity, one has m3/2 = m0 (4) if the cosmo-

logical constant Λ = 0, and the identity of the LSP varies over the (m1/2, m0) plane. We

have recently published an analysis which includes the possibility that the gravitino is the

LSP possibility [8], taking into account the constraints imposed by Big-Bang nucleosynthesis

(BBN) and the cosmic microwave background (CMB) data on decays of the next-to-lightest

sparticle (NSP) into the gravitino, as well as the relic gravitino dark matter density itself.

In this paper, we incorporate this analysis into a unified treatment of the neutralino and

gravitino LSP regions of the (m1/2, m0) planes in VCMSSMs.

We display in Fig. 6 the contours of tan β (solid blue lines) in the (m1/2, m0) planes for

selected values of A, B and the sign of µ. Also shown are the contours where mχ± > 104 GeV

(near-vertical black dashed lines) and mh > 114 GeV (diagonal red dash-dotted lines). The

regions excluded by b → sγ have medium (green) shading, and those where the relic density

of neutralinos lies within the WMAP range 0.094 ≤ Ωχh2 ≤ 0.129 have light (turquoise)

shading. The gravitino LSP and the neutralino LSP regions are separated by dark (chocolate)

solid lines, and the WMAP relic-density strip for neutralinos is shown only above these lines.

The regions disfavoured by gµ − 2 at the 2-σ level are very light (yellow) shaded.

If A has a large negative value, we do not find any consistent solutions to the electroweak

vacuum conditions. This is reflected in panel (a) of Fig. 6, for µ > 0 and A = −1.5, where

there are no solutions above the topmost solid (black) line. The solid (blue) contours of tanβ

rise diagonally from low values of (m1/2, m0) to higher values, with higher values of tanβ

having lower values of m0 for a given value of m1/2. The dash-dotted (red) mh = 114 GeV

13

100 200 300 400 500 600 700 800 900 10000

100

200

300

400

500

600

700

800

m1/2 (GeV)

m0

(GeV

)

tan β = 10, µ > 0, A = 0.75^

mh = 114 GeV

0.10

-0.1 -0.25

-0.5

-1

-2

-5

mt = 178 GeV

100 200 300 400 500 600 700 800 900 10000

100

200

300

400

500

600

700

800tan β = 10, µ < 0, A = 0.75^

m1/2 (GeV)

m0

(GeV

)

mh = 114 GeV

0.25

0.5

2/3

1

2

5

mt = 178 GeV

100 1000 20000

1000

m1/2 (GeV)

m0

(GeV

)

tan β = 35, µ < 0, A = 0.75^

mh = 114 GeV

0.5

2/3

1

2

5

mt = 178 GeV

100 1000 2000 30000

1000

1500

m1/2 (GeV)

m0

(GeV

)

tan β = 50, µ > 0, A = 0.75^

mh = 114 GeV

mt = 178 GeV

0.5

2/3

1

25

0.5

Figure 5: The (m1/2, m0) planes in the CMSSM for the same values of tan β and the sign of

µ as in Fig. 4, but here for A = +0.75.

14

100 200 300 400 500 600 700 800 900 10000

1000

m1/2 (GeV)

m0

(GeV

)A = -1.5; µ > 0^

tan β = 2

5 10

25

mh = 114 GeV

100 200 300 400 500 600 700 800 900 10000

1000

m0

(GeV

)m1/2 (GeV)

tan β =

5

10

15 20

25

A = 0.75; µ > 0^

mh = 114 GeV

100 200 300 400 500 600 700 800 900 10000

1000

m1/2 (GeV)

m0

(GeV

)

A = 3 - √3 ; µ > 0^

tan β =

15

10

10

20

25

mh = 114 GeV

100 200 300 400 500 600 700 800 900 10000

1000

m1/2 (GeV)

m0

(GeV

)

A = 2 ; µ > 0^

tan β = 30

25

20

mh = 114 GeV

Figure 6: Examples of VCMSSM (m1/2, m0) planes with contours of tan β superposed, for

µ > 0 and (a) A = −1.5, (b) A = 0.75, (c) the simplest Polonyi model with A = 3 −√

3,and (d) A = 2.0, all with B = A − 1. In each panel, we show the regions excluded by theLEP lower limits on MSSM particles and those ruled out by b → sγ decay [31] (mediumgreen shading): the regions disfavoured by gµ − 2 are very light (yellow) shaded, bordered bya thin (black) line. The dark (chocolate) solid lines in panels (a, b, c) separate the gravitinoLSP regions (below). Panel (d) exhibits a dark (red) wedge where the LSP is the rapid τ1.The regions favoured by WMAP in the neutralino LSP case have light (turquoise) shading.The dashed (pink) line corresponds to the maximum relic density for the gravitino LSP, andregions allowed by BBN/CMB constraint on NSP decay are light (yellow) shaded.

15

contour rises in a similar way, and regions above and to the left of this contour have mh < 114

GeV and are excluded. In particular, a neutralino LSP is excluded in this case. We exhibit

in this and the other panels a gravitino LSP region, which was not studied in our previous

exploration of VCMSSMs [6]. The relic density is acceptably low only below the dashed

(pink) line. This excludes a supplementary domain of the (m1/2, m0) plane, but the strongest

constraint is provided by the BBN/CMB decay constraint (light, yellow shading), which

requires tanβ >∼ 4.5. In panels (b, c, d), the mh contour rises more vertically, but only in

panel (d) is there any allowed neutralino LSP region. Panel (d) features an excluded dark

(red) shaded wedge where the LSP is the τ1.

When A is increased to 0.75, as seen in panel (b) of Fig. 6, both the tanβ and mh contours

rise more rapidly with m1/2. Again, there is no allowed neutralino LSP region. Within the

gravitino LSP region, the mh and relic density constraints would both be compatible with

tan β >∼ 7.5, but the BBN/CMB decay constraint imposes the stronger constraint that

tan β >∼ 13. It is instructive to compare this figure with Fig. 5a, which both assume that

A = 0.75. The most notable difference is that, here, fixing the gravitino mass to equal m0

excludes the neutralino coannihilation region with B = −0.25 and allows a region of the

(m1/2, m0) plane that would previously have been excluded because the LSP would have

been the τ1.

An analogous pattern is seen in the simplest Polonyi model with A = 3 −√

3 shown in

panel (c) of Fig. 6, where we note that the tan β contours have noticeable curvature. Once

again, the neutralino LSP region is excluded, now by a combination of the Higgs and chargino

mass bounds. At low m0 in the gravitino LSP region, the mh and relic gravitino density

constraints impose tan β >∼ 10 and the BBN/CMB decay constraint imposes tanβ >∼ 14 5.

We consider finally the case A = 2.0 shown in panel (d) of Fig. 6. In this case, there is a

neutralino LSP region in the χ − τ coannihilation strip. Without the gµ − 2 constraint, the

most severe constraint on this region is imposed by b → sγ, requiring tanβ >∼ 25, the mh

constraint being much weaker. Imposing the gµ − 2 constraint requires tanβ >∼ 27. There is

an excluded dark (red) shaded wedge where the LSP is the τ1. Below this appears a gravitino

LSP region with acceptable relic density. Within this region, the mh and BBN/CMB decay

constraints impose tan β >∼ 15, which would be strengthened to tan β >∼ 20 if one took the

gµ − 2 constraint at face value. This is the shaded region in the lower right of panel (d).

We find no consistent solutions for values of A substantially greater than 3 (4) when

µ > 0 (µ < 0), and negative values of A are not allowed when µ < 0. These restrictions arise

5There is also a negative Polonyi solution with A = −3 +√

3, whose (m1/2, m0) plane is qualitativelysimilar to panel (a) of Fig. 6.

16

100 200 300 400 500 600 700 800 900 10000

1000

m1/2 (GeV)

m0

(GeV

)A = 3 - √3 ; µ < 0^

tan

β =

5

mh = 114 GeV

4

6

100 200 300 400 500 600 700 800 900 10000

1000

m1/2 (GeV)m

0 (G

eV)

A = 2 ; µ < 0^

tan β = 10

20 15

5

mh = 114 GeV

Figure 7: As in Fig. 6, but now for µ < 0 and the choices (a) A = 3 −√

3, B = A − 1 and(b) A = 2, B = A − 1 and µ < 0.

from the behavior of the relation between tanβ and B0 discussed earlier. Therefore as A

increases, so does the solution for tan β when B = A − 1. At very large tanβ, there are no

solutions to the RGEs due to a divergence in the bottom-quark Yukawa coupling. For small

A and µ < 0, the solution is driven to excessively small values of tan β, where again there

are no solutions, now due to the divergence in the top Yukawa coupling mentioned earlier.

The same is true when A is large and negative and µ > 0, i.e. for A < −2.5, m0<∼ 500 GeV

for m1/2 ≤ 1000 GeV.

As we see in panel (a) of Fig. 7, only a small area of the (m1/2, m0) plane in the gravitino

LSP region is allowed by the mh constraint in the positive Polonyi case A = 3−√

3 6. This

area would be further restricted if one took the gµ − 2 constraint at face value. At larger

values of A, the allowed region is extended, as exemplified in panel (b) of Fig. 7 for the

case A = 2, where the mh constraint is somewhat weaker. However, in this case the gµ − 2

constraint would have a much more drastic effect.

6The negative Polonyi case is not allowed for µ < 0.

17

6 The µ Problem and the Giudice-Masiero Mechanism

One of the primary motivations in building supersymmetric model is to avoid the the gauge

hierarchy problem, namely that the Higgs mass is of order mZ and much less than the Planck

mass, though not protected by any symmetry of the Standard Model between the GUT scale

and the weak scale. Supersymmetry alleviates this problem via cancellations between contri-

butions to the Higgs mass from fermions and bosons in the same supermultiplet. However,

this scenario begs the question why supersymmetry is broken by soft terms which are as-

sumed to be O(1 TeV). Moreover, there is one other, supersymmetric, parameter which is

required to be small, namely the Higgs mixing parameter µ. One of the most interesting

attempts to explain the smallness of µ is the Giudice-Masiero mechanism [5], in which it

is related to a coupling between observable and hidden sectors, and is of the same order

of magnitude as the soft supersymmetry-breaking parameters. In the simplest realization

of the Giudice-Masiero mechanism with only one hidden superfield, one has the following

relation between A and B, as already mentioned:

B =2A − 3

A − 3, (20)

and∣

∣

∣

∣

µ

m0

∣

∣

∣

∣

=

∣

∣

∣

∣

∣

λA − 3√

3

∣

∣

∣

∣

∣

(21)

where λ is the coupling constant between the hidden superfield and the two Higgs supermul-

tiplets. One should require that λ ∼ O(1) for µ to be the same order of m0.

We display in Fig. 8 some typical (m1/2, m0) planes in the Giudice-Masiero model for

positive µ. As in the previous minimal supergravity VCMSSMs, we find no consistent elec-

troweak solutions for values of A much outside the range of values exhibited. In the examples

shown, there are no solutions above the topmost solid lines in panels (a) for A = 0.6 and (d)

for A = 1.8. For A < −0.6, m0<∼ 150 GeV for m1/2 ≤ 1000 GeV. Similarly for A >∼ 2.6 only

a small corner of the plane admits solutions.

In panel (a) for A = 0.6, corresponding to B = 0.75, there is no allowed region above the

solid (chocolate) gravitino LSP line. Below this line, we see an allowed region for tan β >∼ 22.

However, we also note that the corresponding values of λ are quite large, λ >∼ 5. The

situation is somewhat different for the case A = 0.8, corresponding to B ≃ 0.64, shown in

panel (b) of Fig. 8. In this case, we see that there is a narrow allowed region along the

χ − τ coannihilation strip in the neutralino LSP region for tanβ >∼ 33.5, or >∼ 35 if the

gµ − 2 constraint is taken into account. This region requires λ >∼ 2, which is relatively

18

palatable. At lower m0, there is a disallowed dark (red) wedge where the τ1 is the LSP, and

below that a region where the gravitino is the LSP. The latter contains a domain allowed

by the BBN/CMB decay constraint, that appears for tan β >∼ 18, or >∼ 20 if one includes

the gµ − 2 constraint. However, this region again has λ >∼ 5. Turning now to the Polonyi

case A = 3 −√

3 shown in panel (c) of Fig. 8, corresponding to B ≃ 0.27, we see that there

is no allowed area in the neutralino LSP region above the dark solid (chocolate) line, and

that the allowed region in the gravitino LSP region requires tanβ >∼ 13.5 and again λ >∼ 5.

Similar features are seen in panel (d) for A = 1.8, corresponding to B = −0.5, where the

only allowed area - in the gravitino LSP region - requires even larger values of λ than the

previous cases.

Fig. 9 shows some analogous cases for µ < 0. As before, there are no consistent elec-

troweak vacuum solutions for values of A substantially outside the range of values shown, and

none above the topmost solid lines in panels (a) and (b). Panel (a) is for A = −0.2, corre-

sponding to B ≃ 1.06. It has two narrow strips in the neutralino LSP region that are allowed

if one discards the gµ − 2 constraint, appearing for m1/2>∼ 800 GeV and m0

>∼ 500 GeV

for tan β > 38 and λ < 1.5. Down in the gravitino LSP area, there is a second allowed

region with tan β <∼ 13 and λ >∼ 3. For smaller values of A, the allowed parameter space is

further squeezed. For example, for A = −1 we find m0<∼ 500 GeV for m1/2 ≤ 1000 GeV.

In panel (b) for A = 0.6, corresponding to B = 0.75, the allowed neutralino LSP region has

disappeared, but a gravitino LSP region remains. Similar features are seen in panels (c) and

(d) for A = 1 (B = 0.5) and A = 3−√

3 (B ≃ 0.27), respectively. For A >∼ 2, solutions exist

only in a small portion of the plane.

7 Conclusions

We have discussed in this paper the impacts of the theoretical, experimental and cosmological

constraints on some classes of VCMSSMs, including minimal supergravity models and the

Giudice-Masiero model. We have presented unified treatments of the regions of parameter

space in these models where the LSP is a neutralino or the gravitino.

We have emphasized that the predictions of these models differ significantly from those

of the CMSSM. In particular, the CMSSM is distinct from minimal supergravity: the for-

mer does not necessarily require a fixed relation between the trilinear and bilinear soft

supersymmetry-breaking parameters A, B, nor equality between m0 and m3/2, as required

in minimal supergravity models. The values of B required in generic realizations of the

CMSSM generally bear no relation to the values that would be derived in minimal super-

19

100 200 300 400 500 600 700 800 900 10000

1000

0.094

0.094

0.09

4

0.09

4

0.09

4

0.129

0.129

00.129

0.129

0.12

9

0.129

m1/2 (GeV)

m0

(GeV

)A = 0.6 ; µ > 0^

λ = 2

58

tan β = 40

510

3030

100 200 300 400 500 600 700 800 900 10000

1000

0.094

m1/2 (GeV)m

0 (G

eV)

A = 0.8 ; µ > 0^

tan

β =

58

tan β = 40

30

25

λ = 1

λ = 2

5

1020

100 200 300 400 500 600 700 800 900 10000

1000

1

m1/2 (GeV)

m0

(GeV

)

mh = 114 GeV

λ =

1

λ = 2

5

tan β = 15

10

10

20

1020

50

25

A = 3 - √3 ; µ > 0^

100 200 300 400 500 600 700 800 900 10000

1000

1.9

1.9

1.9 1.9

1.9

2

2

2 2

2

2

5

5

5

5

10

10

25

3057.5

5

5

5

5

5

5

2 2

m1/2 (GeV)

m0

(GeV

)

A = 1.8 ; µ > 0^

tan β

= 2λ = 5 5 10

50

2015

20

Figure 8: Examples of (m1/2, m0) planes in the Giudice-Masiero model, with contours of

tan β superposed, for µ > 0 and (a) A = 0.6, (b) A = 0.8, (c) A = 3 −√

3 and (d) A = 1.8.In each panel, we show the regions excluded by the LEP lower limits on MSSM particles,those ruled out by b → sγ decay [31] (medium, green shading), and those disfavoured bygµ − 2 (very light, yellow shading). As before, a dotted (red) line shows where mχ = mτ andthe gravitino LSP region is bounded by a solid (chocolate) line in panels (a, c, d). The dark(red) wedge in panel (b) has a τ1 LSP and hence is disallowed. Within the gravitino LSPregion, the relic density constraint is indicated by a dashed (pink) line and the BBN/CMBconstraint on NSP decay by light (yellow) shading. The dashed black lines are contours ofthe Giudice-Masiero parameter λ.

20

100 200 300 400 500 600 700 800 900 10000

1000

0.09

4

0.094

0.094

0.09

4

0.09

4

0.12

9

0.129

0.12

9 0.12

9

0.12

9

0.12

9

0.129

0.12

9

0.12

9

m0

(GeV

)

m1/2 (GeV)

A = -0.2 ; µ < 0^

tan β = 30

20

10

5

40

λ = 2

5 10

100 200 300 400 500 600 700 800 900 10000

1000

m0

(GeV

)

m1/2 (GeV)

A = 0.6 ; µ < 0^

mh = 114 GeV tan β = 20

3040

10

15

5

λ = 2

10

5

100 200 300 400 500 600 700 800 900 10000

1000

2

m1/2 (GeV)

m0

(GeV

)

A = 1 ; µ < 0^

mh = 114 GeV

tan β =

10

1520

25

λ = 2

λ = 5

105

100 200 300 400 500 600 700 800 900 10000

1000A = 3 - √3 ; µ < 0^

mh = 114 GeV

m0

(GeV

)

m1/2 (GeV)

tan

β =

5

λ = 2

λ = 5

λ = 10

Figure 9: Same as Fig. 8, but for µ < 0 and (a) A = −0.2, (b) A = 0.6, (c) A = 1, and (d)A = 3 −

√3.

21

gravity models.

In addition to minimal supergravity models, we have discussed the simplest variant of the

Giudice-Masiero model, which makes a brave attempt to provide a framework for calculating

the Higgs-mixing superpotential term µ.

There are a couple of striking features of these specific analyses that we note. One is that

the range of A is often very restricted: beyond this range, it is impossible to find consistent

solutions to the electroweak vacuum conditions.

A second observation is that, in both minimal supergravity and the Giudice-Masiero

model, a neutralino LSP is completely excluded in many instances, and the gravitino LSP

regions are generally much more extensive than the neutralino LSP regions. To some extent,

this was to be expected, since we impose the cosmological dark matter density and NSP decay

constraints on gravitino dark matter as one-sided upper limits, rather than as narrow WMAP

ranges as for the dark matter density constraint on neutralino dark matter. This is because,

in the case of gravitino dark matter, the narrow range could be reached by postulating

thermal gravitino production with a suitable reheating temperature [32]. Of course, in

either the neutralino or gravitino case, one could always postulate a supplementary source of

cold dark matter. In the case of neutralino dark matter, this possibility would broaden the

WMAP density strip down to the mχ = mτ boundary. However, the gravitino dark matter

region would still, for many choices of the other supersymmetric parameters, occupy a larger

area of the (m1/2, m0) plane.

In any complete supersymmetric theory, one expects some relations between supersym-

metry breaking parameters, perhaps of the type discussed here. In this case, some VCMSSM

should be responsible for the low-energy sparticle spectrum. However, we do not yet know

what specific constraints are handed down from the unification or string scales. As we have

emphasized in this paper, the predictions in such models may differ greatly from those of the

more relaxed CMSSM and, a priori, those of a more general MSSM. Analogous differences

are also to be expected in the predicted cross sections for direct and indirect searches for

supersymmetric dark matter, a topic we will consider elsewhere.

Acknowledgments

The work of K.A.O., Y.S., and V.C.S. was supported in part by DOE grant DE–FG02–

94ER–40823. Y.S. would like to thank D.A. Demir for asking about the Giudice-Maisero

mechanism in this context, and for helpful conversations.

22

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